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Relative Error of Scaled Poisson Approximation via Stein's Method

Yue Tan, Yingdong Lu, Cathy Xia

Abstract

We study the accuracy of a scaled Poisson approximation to the weighted sum of independent Poisson random variables, focusing on in particular the relative error of the tail distribution. A bound on the relative approximation error is established using a modified Stein-Chen method.

Relative Error of Scaled Poisson Approximation via Stein's Method

Abstract

We study the accuracy of a scaled Poisson approximation to the weighted sum of independent Poisson random variables, focusing on in particular the relative error of the tail distribution. A bound on the relative approximation error is established using a modified Stein-Chen method.

Paper Structure

This paper contains 20 sections, 23 theorems, 115 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results and Implications
  3. Scaled Poisson Approximation
  4. Stein's Method for Scaled Poisson Random Variable
  5. Main Results on Relative Errors for Scaled Poisson Approximation
  6. On a conjecture in liu_xia_2020
  7. Proof of Theorem \ref{['thm:main']}
  8. Properties of the Solution to the Stein Equation
  9. Some Basic Properties of $f_h$
  10. The Case of $w \ge my$
  11. The Case of $w< my$
  12. Conclusions
  13. Proof of Lemma \ref{['lem:SteinEqnNew']}
  14. Proof of Lemma \ref{['lem:0']}
  15. Proof of Lemma \ref{['lem:Poisson_sum_decom']}
  16. ...and 5 more sections

Key Result

Lemma 2.1

The Stein's operator for $\hat{A}_{\lambda}$ is That is, $E(\mathcal{A}f)(\hat{A}_{\lambda})=0$ for all real-valued functions $f$ defined on $Z_{\ge 0}$.

Figures (1)

  • Figure 1: Comparison of ratio+, ratio-, ratio 1 and ratio 2 for $y \in [25, 200]$ and $k = \lfloor\frac{y}{m}\rfloor$ with $m=9$.

Theorems & Definitions (30)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.1
  • Corollary 1
  • proof
  • Theorem 2.2: Main Result
  • proof
  • Remark 2.1
  • ...and 20 more